basic hadronic SU(3) model generating a critical end point in a hadronic model revisited including quark degrees of freedom phase diagram – the QH model excluded volume corrections, phase transition J. Steinheimer, V. Dexheimer, P. Rau, H. Stöcker, SWS FIAS; Goethe University, Frankfurt OUTLINE Hot and dense matter in quark-hadron models ICPAQGP, Goa 2010

A) SU(3) interaction ~ Tr [ B, M ]  B, ( Tr B B ) Tr M B) meson interactions ~ V(M) =  0  0 =  0  0 C) chiral symmetry  m  = m K = 0 explicit breaking ~ Tr [ c  ] (  m q q q )  light pseudoscalars, breaking of SU(3) _ _ hadronic model based on non-linear realization of chiral symmetry degrees of freedomSU(3) multiplets:  ~  0 ~ baryons (n,Λ, Σ, Ξ) scalars ( , ,  0 ) vectors (ω, ρ, φ), pseudoscalars, glueball field χ _ _ _ _ _ _

fit parameters to hadron masses ’’  mesons Model can reproduce hadron spectra via dynamical mass generation p,n     K          K* ** **  

Lagrangian (in mean-field approximation) L = L BS + L BV + L V + L S + L SB baryon-scalars: L BS = -  B i (g i   + g i   + g i   ) B i L BV = -  B i (g i   + g i   + g i   ) B i baryon-vectors: meson interactions: L BS = k 1 (  2 +  2 +  2 ) 2 + k 2 /2 (   4 +   2  2 ) + k 3   2  - k 4  4 -  4 ln  /  0 +   4 ln [(  2 -  2 )  / (  0 2  0 )] explicit symmetry breaking: L SB = c 1  + c 2  _ _ L V = g 4 (  4 +  4 +  4 + β  2  2 ) / / / I I

parameter fit to known nuclear binding energies and hadron masses 2d calculation of all measured (~ 800) even-even nuclei error in energy  (A  50) ~ 0.21 % (NL3: 0.25 %)  (A  100) ~ 0.14 % (NL3: 0.16 %) good charge radii  r ch ~ 0.5 % (+ LS splittings) SWS, Phys. Rev. C66, (2002) relativistic nuclear structure models + correct binding energies of hypernuclei compressibility ~ 223 MeV asymmetry energy ~ 31.9 MeV binding energy E/A ~ MeV saturation (  B ) 0 ~.16/fm 3 phenomenology: MeV MeV Nuclear Matter and Nuclei

phase transition compared to lattice simulations heavy states/resonance spectrum is effectively described by single (degenerate) resonance with adjustable couplings reproduction of LQCD phase diagram, especially T c, μ c + successful description of nuclear matter saturation phase transition becomes first-order for degenerate baryon octet ~ N f = 3 with T c ~ 185 MeV T c ~ 180 MeV µ c ~ 110 MeV D. Zschiesche et al. JPhysG 34, 1665 (2007)

Isentropes, UrQMD and hydro evolution J. Steinheimer et al. PRC77, (2008) lines of constant entropy per baryon, i.e. perfect fluid expansion E/A = 5, 10, 40, 100, 160 GeV E/A = 160 GeV goes through endpoint

P. Rau, J. Steinheimer, SWS, in preparation Including higher resonances explicitly Add resonances up to 2.2 GeV. Couple them like the lowest-lying baryons

Include modified distribution functions for quarks/antiquarks Following the parametrization used in PNJL calculations The switch between the degrees of freedom is triggered by excluded volume corrections thermodynamically consistent - D. H. Rischke et al., Z. Phys. C 51, 485 (1991) J. Cleymans et al., Phys. Scripta 84, 277 (1993) U = - ½ a(T) ΦΦ* + b(T) ln[1 – 6 ΦΦ* + 4 (ΦΦ*) 3 – 3 (ΦΦ*) 2 ] a(T) = a 0 T 4 + a 1 T 0 T 3 + a 2 T 0 2 T 2, b(T) = b 3 T 0 3 T χ = χ o (1 - ΦΦ* /2) V q = 0 V h = v V m = v / 8 µ i = µ i – v i P ~ different approach – hadrons, quarks, Polyakov loop and excluded volume e = e / (1+ Σ v i ρ i ) ~~ Steinheimer,SWS,Stöcker hep-ph/ * *

quark, meson, baryon densities at µ = 0 natural mixed phase, quarks dominate beyond 1.5 T c densities of baryon, mesons and quarks Energy density and pressure compared to lattice simulations ρ

Interaction measure e – 3p Temperature dependence of chiral condensate and Polaykov loop at µ = 0 lattice data taken from Bazavov et al. PRD 80, (2009) speed of sound shows a pronounced dip around T c !

Lattice comparison of expansion coefficients as function of T expansion coefficients lattice data from Cheng et al., PRD 79, (2009) lattice results Steinheimer,SWS,Stöcker hep-ph: suppression factor peaks

Φ Dependence of chiral condensate on µ, T Lines mark maximum in T derivative σ Separate transitions in scalar field and Polyakov loop variable

Φ Dependence of Polyakov loop on µ, T Lines mark maximum in T derivative Separate transitions in scalar field and Polyakov loop variable σ

Susceptibilitiy c 2 in PNJL and QHM for different quark vector interactions Steinheimer,SWS, hepph/ g qω = g nω /3 g qω = 0 PNJL QH At least for µ = 0 – small quark vector repulsion

σ Φ

UrQMD/Hydro hybrid simulation of a Pb-Pb collision at 40 GeV/A red regions show the areas dominated by quarks

SUMMARY general hadronic model as starting point works well with basic vacuum properties, nuclear matter, nuclei, … phase diagram with critical end point via resonances implement EOS in combined molecular dynamics/ hydro simulations quarks included using effective deconfinement field implementing excluded volume term, natural switch of d.o.f. If you want to do some lattice/quark calcs, grab your iPhone -> Physics to Go! Part 3

order parameter of the phase transition confined phase deconfined phase effective potential for Polyakov loop, fit to lattice data quarks couple to mean fields via g σ, g ω connect hadronic and quark degrees of freedom minimize grand canonical potential baryonic and quark mass shift δ m B ~ f(Φ) δ m q ~ f(1-Φ) V. Dexheimer, SWS, PRC (2010) Ratti et al. PRD (2006) Fukushima, PLB 591, 277 (2004) U = ½ a(T) ΦΦ* + b(T) ln[1 – 6 ΦΦ* + 4 (ΦΦ*) 3 – 3 (ΦΦ*) 2 ] a(T) = a 0 T 4 + a 1 µ 4 + a 2 µ 2 T 2 q q

hybrid hadron-quark model critical endpoint tuned to lattice results Phase Diagram for HQM model µ c = 360 MeV T c = 166 MeV µ c. = 1370 MeV ρ c ~ 4 ρ o V. Dexheimer, SWS, PRC (2010)

C s ph ~ ¼ C s,ideal isentropic expansion overlap initial conditions E lab = 5, 10, 40, 100, 160 AGeV averaged C s significantly higher than 0.2

important reality check compressibility ~ 223 MeV asymmetry energy ~ 31.9 MeV equation of state E/A (  ) asymmetry energy E/A (  p -  n ) nuclear matter properties at saturation density binding energy E/A ~ MeV saturation (  B ) 0 ~.16/fm 3 phenomenology: MeV MeV + good description of finite nuclei / hypernuclei SWS, Phys. Rev. C66,

subtracted condensate and polyakov loop different lattice groups and actions From Borsanyi et al., arxiv:1005:3508 [hep-lat]

fsfs If you want it exotic … follow star calcs by J. Schaffner et al., PRL89, (2002) E/A-m N additional coupling g 2 of hyperons to strange scalar field g 2 = 0 g 2 = 2 g 2 = 4 g 2 = 6 barrier at f s ~ 0.4 – simple time evolution including π, K evaporation (E/A = 40 GeV) C. Greiner et al., PRD38, 2797 (1988) with evaporation

Temperature distribution from UrQMD simulation as initial state for (3d+1) hydro calculation dip in c s is smeared out Speed of sound - (weighted) average over space-time evolution initial temperature distribution

Hypernuclei -  single-particle energies Model and experiment agree well Nuclear matter

Evolution of the collision system E lab ≈ 5-10 AGeV sufficient to overshoot phase border, AGeV around endpoint

amount of volume scanning the critical endpoint (lattice)

Mass-radius relation using Maxwell/Gibbs construction Gibbs construction allows for quarks in the neutron star mixed phase in the inner 2 km core of the star V. Dexheimer, SWS, PRC (2010) R. Negreiros, V. Dexheimer, SWS, PRC, astro- ph: